Flip a fair coin until you get the first "head". Let X represent the number of flips before the first head appears. Calculate E[X].
So I solved this problem and you get a power series:
$E[X] = 1*0.5 + 2*0.5^2 + 3*0.5^3+ ...$
This is basically of the form $\sum\limits_{i=0}^{\infty}x_i(0.5)^i.$
I flipped open my calculus book to review how to solve this series but I didn't find anything on how to calculate the limiting value for a power series, just the radius of convergence.
I see for a sum of infinite geometric series, the value is:
$S_n = a+ ax + ax^2 + ... + ax^n + ... = \cfrac{a}{1-x}$ for |x| < 1.
Can someone please tell me the general approach if there is one for a power series?
Thank you in advance.